Algebro-geometric integration of the modified Belov-Chaltikian lattice hierarchy. (English. Russian original) Zbl 1429.37041
Theor. Math. Phys. 199, No. 2, 675-694 (2019); translation from Teor. Mat. Fiz. 199, No. 2, 235-256 (2019).
Summary: Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov-Chaltikian lattice hierarchy associated with a discrete \(3 \times 3\) matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve \(K_{m -2}\) of arithmetic genus \(m- 2\). We study the asymptotic properties of the Baker-Akhiezer function and the algebraic function carrying the data of the divisor near \(P_{\infty_1}\), \(P_{\infty_2}\), \(P_{\infty_3}\), and \(P_0\) on \(K_{m -2} \). Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker-Akhiezer function, and, in particular, solutions of the entire modified Belov-Chaltikian lattice hierarchy.
MSC:
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 14H70 | Relationships between algebraic curves and integrable systems |
Keywords:
Belov-Chaltikian lattice hierarchy; trigonal curve; quasiperiodic solution; Baker-Akhiezer functionReferences:
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